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There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book Isoperimetric Inequalties in Mathematical Physics) which is as follows:

"Of all $n$-gons of a fixed area, the regular $n$-gon minimizes the first Dirichlet eigenvalue."

Surprisingly, this is still open (to my knowledge) for the general case. The only settled cases are the triangles and the quadrilaterals (see Henrot's survey). Is there any progress on the general case?

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    $\begingroup$ Is the regular $n$-gon even proved to be a local minimum for $\lambda_1$? The survey you cite doesn't cite such a result in the "case of polygons" sections (3.2, page 5). $\endgroup$ Apr 10, 2016 at 15:19
  • $\begingroup$ @NoamD.Elkies As far as I know, the Polya's proof is the only one out there. I have not seen any proof for even "local"minimality. I thought I had a good idea how to approach this problem but sadly my method is only good for quadrilaterals. $\endgroup$
    – BigM
    Apr 10, 2016 at 16:08
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    $\begingroup$ Is this conjecture also connected in any way to the Selberg eigenvalue conjecture, or they just happen to share the word "eigenvalue"? $\endgroup$
    – Suvrit
    Apr 10, 2016 at 16:48
  • $\begingroup$ At Survit, I looked at the link(Peter Sarnak's) paper on Selberg's conjecture.That is not in my realm however to my understanding Selberg's inequality is analog of general isopremeteric inequality stating among regions of same area circle(balls for higher dimensions)maximizes the first eigenvalue. M.Ruzhansky(Imperial College London) has some nice results pertaining iso. inequalities for Lie groups and connected Riemannian manifolds. $\endgroup$
    – BigM
    Apr 10, 2016 at 20:59
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    $\begingroup$ Some recent numerical computations suggest that regular polygons are indeed minimizers of the first eigenvalue under area constraint: lama.univ-savoie.fr/~bogosel/faber_krahn_polygons.html I do not know any proof of the fact that regular polygons are local minimizers. It is possible to prove that they are critical points, i.e. the derivative of the first eigenvalue of a regular polygon is zero with respect to every perturbation of the vertices. $\endgroup$ Apr 30, 2016 at 21:43

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I'm pretty sure that this is still open for $n$-gons (with $n\geq 5$). As far as I know, basically no progress has been made since the original proofs for triangles/quadrilaterals.

There have been some numerics as well as some refined inequalities for triangles. This article might point you towards some of these results.

Interestingly, a related conjecture of Pólya & Szegő is resolved "the regular $n$-gon has least logarithmic capacity among $n$-gons of a fixed area" http://www.ams.org/mathscinet-getitem?mr=2052355.

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    $\begingroup$ I remember Solynin gave a talk at our school a few years ago.Back then I wasn't interested in the topic and didnt really pay much attention.Well my loss. At any rate, I glanced at your link to their Annals paper and have to say that their result is very neat. $\endgroup$
    – BigM
    Apr 11, 2016 at 4:00

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